The primitivity of operators in the algebra of binary relations under conjunctions of containments

Abstract

Abstract-The algebra of binary relations provides union and composition as basic operators, with the empty set as neutral element for union and the identity relation as neutral element for composition. The basic algebra can be enriched with additional features. We consider the diversity relation, the full relation, intersection, set difference, projection, coprojection, converse, and transitive closure. It is customary to express boolean queries on binary relational structures as finite conjunctions of containments. We investigate which features are primitive in this setting, in the sense that omitting the feature would allow strictly less boolean queries to be expressible. Our main result is that, modulo a finite list of elementary interdependencies among the features, every feature is indeed primitive. I. INTRODUCTIONThe algebra of binary relations (aka the calculus of relations) was created by De Morgan, Peirce, and Schröder, and popularized and further developed by Tarski and his collaborators [1]- [3]. These developments gave rise to the rich field of relation algebras [4]- [6]. In the present paper, however, we are focusing on the algebra of binary relations as a language for expressing properties of binary relational structures. This focus comes very naturally in the context of query languages for graph data. Indeed, a binary relational structure really is a directed graph, with the different binary relation names playing the role of edge labels. Not surprisingly, the algebra of binary relations lies at the basis of query languages for graph databases [7]-[12].But also more fundamentally, the algebra of binary relations originated in the desire to have a principled language for expressing properties (axiomatizing classes) of relational structures. (In this respect, the algebra of binary relations actually predates first-order logic [13].) This paper is part of our ongoing work [9], [14]-[17] to understand the expressive power, in particular the primitivity or interdependencies, among different operators considered in the realm of binary relational algebra. Since we include projection, coprojection, transitive closure, intersection, and converse, our investigation is also relevant to propositional dynamic logic, multi-dimensional modal logic [18] and to the relational interpretation of Kleene algebras with tests, and of Kleene allegories [19], [20].There are indeed many different operators that have been considered. Our choice of operators is a natural one and motivated by the applications to graph database query languages. We always start from union ∪ and composition •,

A framework for comparing query languages in their ability to express boolean queries

“…2. The proof of Proposition VI.1 in [27] shows that. Lemma 5 ([27]) Any boolean query in NoDiff ⊆ cannot be true on B and false on…”

“…Let id 1 be the graph that consists of a single self-loop. The proof of Proposition IX.1 in [27] shows that id 1 and K 3 are indistinguishable in {all, −1 , π, ∩, + } ⊆ . As a direct corollary, we obtain that di = ∅ ("the graph has at least two nodes") is not in {all, −1 , π, ∩, + } ⊆ .…”

“…Although the assumption of a basic language is a point of discussion, it can be argued that our choice of basic language is not unreasonable [27].…”

“…Dually, one may investigate how different query languages compare for a given modality. In the context of navigational graph query languages, we have already done this research [29,27].…”

“…Previous work. In our previous work we have compared the expressive power of fragments of the algebra of binary relations under the nonemptiness modality [16,15,29] and under the containment modality [27]. The present paper is complementary in that it emphasizes the comparison of different boolean query modalities, for fixed fragments or across fragments.…”